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A theory of relative extensions for subalgebras and submodules. (English) Zbl 0605.13010
Relative extensions of algebras and submodules are studied. These extensions appear naturally in complex analytic deformation theory. Let $$(A,B)$$ and $$(G,f)$$ be two pairs, where $$A\subset B$$ is a subalgebra over the commutative ring $$K$$, $$F$$ is a $$B$$-module, $$G\subset F$$ is a sub-$$A$$-module. Consider all relative extensions $\begin{matrix} 0 & \to & F \to \tilde B & \to & B &\to 0 \\ && \cup && \cup && \cup \\ 0 & \to & G & \to & \tilde A & \to & A & \to & 0 \end{matrix}$ such that both rows are ($$K$$-split) extensions. Let $$\text{Ex}(A,B;G,F)$$ denote the class of all relative extensions. The equivalence relation may be established on $$\text{Ex}(A,B;G,F)$$. In this way we obtain a set $$\text{Ex}(A,B;G,F)$$ of equivalence classes, which endowed with two operations: $$K\times \text{Ex}(\cdot)\to \text{Ex}(\cdot),$$ $$\text{Ex}(\cdot)\times \text{Ex}(\cdot)\to \text{Ex}(\cdot)$$ becomes a $$K$$-module.
Proposition 1. $$\text{Ex}(A,B;G,F)$$ is isomorphic to the $$K$$-module $[C^ 1(A,F/G) \times_{Z^ 2(A,F/G)} Z^ 2(B,F)]/C^ 1(B,F).$ The notations according to S. MacLane [Homology. Berlin etc.: Springer-Verlag (1963; Zbl 0133.26502); 3rd edition (1975; Zbl 0328.18009)] are used here. If $$K$$ is a field, the vector space $$\text{Ex}(A,B;G,F)$$ is isomorphic to $$Z^ 2(A,B;G,F)/C^ 1(A,B;G,F)$$. Using this isomorphism, the author establishes some long exact sequences of relative derivation and extension modules which interlock in an interesting way.
Reviewer: V. Sharko
MSC:
 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13B02 Extension theory of commutative rings 18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
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References:
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